IJMMS 26:11 (2001) 707–712
PII. S0161171201006299
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
ON FUZZY POINTS IN SEMIGROUPS
KYUNG HO KIM
(Received 4 December 2000)
Abstract. We consider the semigroup S of the fuzzy points of a semigroup S, and discuss
the relation between the fuzzy interior ideals and the subsets of S in an (intra-regular)
semigroup S.
2000 Mathematics Subject Classification. 03E72, 20M12.
1. Introduction. After the introduction of the concept of fuzzy sets by Zadeh [8],
several researches were conducted on the generalizations of the notion of fuzzy sets.
Pu and Liu [5] introduced the notion of fuzzy points. In [6, 7, 8], authors characterized
fuzzy ideals as fuzzy points of semigroups. In [1, 2, 3], Kuroki discussed the properties
of fuzzy ideals and fuzzy bi-ideals in a semigroup and a regular semigroup. In this
paper, we consider the semigroup S of the fuzzy points of a semigroup S, and discuss
the relation between the fuzzy interior ideals and the subsets of S in an (intra-regular)
semigroup S.
2. Preliminaries. Let S be a semigroup with a binary operation “·”. A nonempty
subset A of S is called a subsemigroup of S if A2 ⊆ A, a left (resp., right ) ideal of S if
SA ⊆ A (resp., AS ⊆ A), and a two-sided ideal (or simply ideal) of S if A is both a left
and a right ideal of S. A subsemigroup A of S is called a bi-ideal of S if ASA ⊆ A. Let S
be a semigroup. A nonempty subset A of S is called an interior ideal of S if SAS ⊆ A.
A function f from a set X to [0, 1] is called a fuzzy subset of X. The set {x ∈ X |
f (x) > 0} is called the support, denoted by supp f , of f . The closed interval [0, 1] is a
complete lattice with two binary operations “∨” and “∧”, where α∨β = sup{α, β} and
α ∧ β = inf{α, β} for each α, β ∈ [0, 1]. For any α ∈ (0, 1] and x ∈ X, a fuzzy subset
xα of X is called a fuzzy point in X if
xα (y) =

α
if x = y,
0
otherwise,
(2.1)
for each y ∈ X. If f is a fuzzy subset of X, then a fuzzy point xα is said to be contained
in f , denoted by xα ∈ f , if α ≤ f (x). It is clear that xα ∈ f for some α ∈ (0, 1] if and
only if x ∈ supp f .
A fuzzy subset f of a semigroup S is called a fuzzy subsemigroup of S if
f (xy) ≥ f (x) ∧ f (y),
for all x, y ∈ S, a fuzzy left (resp., right ) ideal of S if
(2.2)
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KYUNG HO KIM
f (xy) ≥ f (y) resp., f (xy) ≥ f (x) ,
(2.3)
for all x, y ∈ S, and a fuzzy ideal of S if f is both a fuzzy left and a fuzzy right
ideal of S. It is clear that f is a fuzzy ideal of a semigroup S if and only if f (xy) ≥
f (x) ∨ f (y) for all x, y ∈ S, and that every fuzzy left (right, two-sided) ideal of S is a
fuzzy subsemigroup of S.
3. Interior ideals of fuzzy points. Let Ᏺ(S) be the set of all fuzzy subsets of a
semigroup S. For each f , g ∈ Ᏺ(S), the product of f and g is a fuzzy subset f ◦ g
defined as follows:
  f (y) ∧ g(z) if x = yz (y, z ∈ S),
(f ◦ g)(x) =
0
otherwise,
(3.1)
for each x ∈ S. It is clear that (f ◦g)◦h = f ◦(g ◦h), and that if f ⊆ g, then f ◦h ⊆ g ◦h
and h ◦ f ⊆ h ◦ g for any f , g, and h ∈ Ᏺ(S). Thus Ᏺ(S) is a semigroup with the
product “◦”.
Let S be the set of all fuzzy points in a semigroup S. Then xα ◦ yβ = (xy)α∧β ∈ S
and xα ◦ (yβ ◦ zγ ) = (xyz)α∧β∧γ = (xα ◦ yβ ) ◦ zγ for any xα , yβ , and zγ ∈ S. Thus S is
a subsemigroup of Ᏺ(S).
For any f ∈ Ᏺ(S), f denotes the set of all fuzzy points contained in f , that is,
f = {xα ∈ S | f (x) ≥ α}. If xα ∈ S, then α > 0.
For any A, B ⊆ S, we define the product of two sets A and B as A ◦ B = {xα ◦ yβ |
xα ∈ A, yβ ∈ B}.
Lemma 3.1 (see [7, Lemma 4.1]). Let f be a nonzero fuzzy subset of a semigroup S.
Then the following conditions are equivalent:
(1) f is a fuzzy left (right, two-sided) ideal of S.
(2) f is a left (right, two-sided) ideal of S.
Lemma 3.2 (see [7, Lemma 4.2]). Let f and g be two fuzzy subsets of a semigroup S.
Then
(1) f ∪ g = f ∪ g.
(2) f ∩ g = f ∩ g.
(3) f ◦ g ⊇ f ◦ g.
A fuzzy subsemigroup f of a semigroup S is called a fuzzy interior ideal of S if
f (xay) ≥ f (a) for all x, a, y ∈ S.
Lemma 3.3. Let f be a nonzero fuzzy subset of a semigroup S. Then the following
conditions are equivalent:
(1) f is a fuzzy interior ideal of S.
(2) f is an interior ideal of S.
Proof. Let f be a fuzzy interior ideal of S, and let xα , zγ ∈ S and yβ ∈ f . Then
since α > 0, γ > 0, and 0 < β ≤ f (y), we have
0 < α ∧ β ∧ γ ≤ α ∧ f (y) ∧ γ ≤ f (y) ≤ f (xyz).
(3.2)
ON FUZZY POINTS IN SEMIGROUPS
709
Hence xα ◦ yβ ◦ zγ = (xyz)α∧β∧γ ∈ f . This implies that S ◦ f ◦ S ⊆ f , thus f is an
interior ideal of S. Conversely, suppose that f is an interior ideal of S. Let x, y, z ∈ S.
If f (y) = 0, then f (y) = 0 ≤ f (xyz). If f (y) ≠ 0, then yf (y) ∈ f and xf (y) , zf (y) ∈ S.
Since f is an interior ideal of S, we have
(xyz)f (y) = (xyz)f (y)∧f (y)∧f (y) = xf (y) ◦ yf (y) ◦ zf (y) ∈ f .
(3.3)
This implies that f (xyz) ≥ f (y), and hence f is a fuzzy interior ideal of S.
It is clear that any ideal of a semigroup S is an interior ideal of S. It is also clear
that any fuzzy ideal of S is a fuzzy interior ideal of S. A semigroup S is called regular
if, for each element a of S, there exists an element x in S such that a = axa.
Theorem 3.4. Let f be any fuzzy set in a regular semigroup S. Then the following
conditions are equivalent:
(1) f is a fuzzy right (resp., left) ideal of S.
(2) f is an interior ideal of S.
Proof. It suffices to show that (2) implies (1). Assume that (2) holds. Let x be any
element in S. Then since S is regular, there exists element a in S such that x = xax.
If f (x) = 0, f (x) = 0 ≤ f (xy). If f (x) ≠ 0, then xf (x) ∈ f and yf (x) ∈ S. Since f is
an interior ideal of S, we have
(xy)f (x) = (xaxy)f (x)
= (xa)xy f (x)∧f (x)∧f (x)
(3.4)
= (xa)f (x) ◦ xf (x) ◦ yf (x) ∈ f .
This implies that f (xy) ≥ f (x), and hence f is a fuzzy right ideal of S.
Theorem 3.5 (see [7, Theorem 3.3]). Let S be a semigroup. If for a fixed α ∈ (0, 1],
fα : S → S is a function defined by fα (x) = xα , then fα is a one-to-one homomorphism
of semigroups.
From Theorem 3.5, we can consider S as an extension of a semigroup S.
Let f be a fuzzy subset of a semigroup S. If ᏾f is the subset of S × S given as
following:
᏾f = xα , xα | xα ∈ f ∪ xα , xβ | xα , xβ ∈ f ,
(3.5)
then the set ᏾f is an equivalence relation on S. We can consider the quotient set S/᏾f ,
with the equivalence classes x α for each x ∈ S. We will denote the subset {x α | xα ∈ f }
of S/᏾f by E(f ). If x α ∈ E(f ), then x α = x f (x) = {xλ | 0 < λ ≤ f (x)}. If x α ∈ E(f ),
then x α = {xα } (singleton set).
Let f be a fuzzy subsemigroup of S. If the product “∗” on E(f ) is defined by
x α ∗ y β = (xy)α∧β for each x α , y β ∈ E(f ), then E(f ) is a semigroup under the operation “∗”.
Theorem 3.6. Let f be a fuzzy interior ideal of S. Then E(f ) is an interior ideal
of (S/᏾f , ∗).
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KYUNG HO KIM
Proof. Let x α , y β ∈ S/᏾f and aγ ∈ E(f ). Then since xα , yβ ∈ S, aγ ∈ f and f is an
interior ideal of S, (xay)α∧γ∧β = xα ◦aγ ◦yβ ∈ f . Hence x α ∗aγ ∗y β = (xay)α∧γ∧β ∈
E(f ). It follows that E(f ) is an interior ideal of S/᏾f .
A semigroup S is called intra-regular if, for each element a of S, there exists elements x and y in S such that a = xa2 y.
Theorem 3.7. A semigroup S is intra-regular if and only if the semigroup S is intraregular.
Proof. Let aα ∈ S. Then since S is intra-regular and a ∈ S, there exist x, y in S
such that a = xa2 y. Thus xα ∈ S and yα ∈ S and
xα ◦ aα ◦ aα ◦ yα = xα ◦ a2 α ◦ yα = xa2 y α = aα .
(3.6)
Hence S is intra-regular. Conversely, let S be intra-regular and a ∈ S. Then for any
α ∈ (0, 1], there exist elements xβ , yδ ∈ S such that
aα = xβ ◦ aα ◦ aα ◦ yδ = xa2 y β∧α∧δ .
(3.7)
This implies that a = xa2 y and x, y ∈ S.
Theorem 3.8. For a fuzzy set f of an intra-regular semigroup S the following conditions are equivalent:
(1) f is a right (resp., left) ideal of S.
(2) f is an interior ideal of S.
Proof. It is clear that (1) implies (2). Assume that (2) holds. Let x, y be any elements
in S. Then since S is intra-regular, there exist elements a, b in S such that x = ax 2 b.
If f (x) = 0, f (x) = 0 ≤ f (xy). If f (x) ≠ 0, then xf (x) ∈ f and yf (x) ∈ S. Since f is
an interior ideal of S, we have
(xy)f (x) = ax 2 by f (x)
= (ax)x(by) f (x)∧f (x)∧f (x)
(3.8)
= (ax)f (x) ◦ xf (x) ◦ (by)f (x) ∈ f .
This implies that f (xy) ≥ f (x), and hence f is a fuzzy right ideal of S.
Lemma 3.9 (see [3, Lemma 4.1]). For a semigroup S, the following conditions are
equivalent:
(1) S is intra-regular.
(2) L ∩ R ⊂ LR holds for every left ideal L and right ideal R of S.
Lemma 3.10 (see [3, Lemma 4.2]). For a semigroup S, the following conditions are
equivalent:
(1) S is intra-regular.
(2) f ∩ g ⊂ g ◦ f holds for every fuzzy right ideal f and fuzzy left ideal g of S.
Theorem 3.11. For a semigroup S, the following conditions are equivalent:
(1) S is intra-regular.
(2) f ∩ g ⊂ g ◦ f for every fuzzy right ideal f and every fuzzy left ideal g of S.
ON FUZZY POINTS IN SEMIGROUPS
711
Proof. Let f be a fuzzy right ideal and g a left ideal of S. Since S is intra-regular,
f is a right ideal, and g is a left ideal of S, f ∩ g ⊂ g ◦ f by Lemma 3.9.
Conversely, let f be a fuzzy right ideal and g a fuzzy left ideal of S. Let x ∈ S. If
f (x) = 0 or g(x) = 0, then
0 = f (x) ∧ g(x) ⊆ (g ◦ f )(x).
(3.9)
If f (x) ≠ 0 and g(x) ≠ 0, then xf (x)∧g(x) ∈ f and xf (x)∧g(x) ∈ g. Hence
xf (x)∧g(x) ∈ f ∩ g ⊂ g ◦ f ⊆ g ◦ f .
(3.10)
It follows that f (x) ∧ g(x) ⊆ (g ◦ f )(x). Hence (f ∩ g)(x) = f (x) ∧ g(x) ⊆ (g ◦ f )(x)
for all x ∈ S and f ∩ g ⊂ g ◦ f . By Lemma 3.10, S is intra-regular.
Lemma 3.12 (see [4, Lemma 4.3]). For a semigroup S the following conditions are
equivalent:
(1) S is both regular and intra-regular.
(2) B 2 = B for every bi-ideal B of S.
(3) A ∩ B ⊂ AB ∩ BA for all bi-ideals A and B of S.
(4) B ∩ L ⊂ BL ∩ LB for every bi-ideal B and every left ideal L of S.
(5) B ∩ R ⊂ BR ∩ RB for every bi-ideal B and every right ideal R of S.
(6) L ∩ R ⊂ LR ∩ RL for every right ideal R and every left ideal L of S.
A fuzzy subsemigroup f of S is called a fuzzy bi-ideal of S if f (xyz) ≥ f (x)∧f (z)
for all x, y and z ∈ S.
Corollary 3.13. For a semigroup S the following conditions are equivalent:
(1) S is both regular and intra-regular.
(2) f ◦ f = f for every fuzzy bi-ideal f of S.
(3) f ∩ g ⊂ f ◦ g ∩ g ◦ f for all fuzzy bi-ideals f and g of S.
(4) f ∩ g ⊂ f ◦ g ∩ g ◦ f for every fuzzy bi-ideal f and every fuzzy left ideal g of S.
(5) f ∩ g ⊂ f ◦ g ∩ g ◦ f for every fuzzy bi-ideal f and every fuzzy right ideal g of S.
(6) f ∩g ⊂ f ◦g ∩g ◦f for every fuzzy right ideal f and every fuzzy left ideal g of S.
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Kyung Ho Kim: Department of Mathematics, Chungju National University, Chungju
380-702, Korea
E-mail address: ghkim@gukwon.chungju.ac.kr